Question

Given the equation f(x)=x^3/(x^2-1), give the x value where the

absolute maximum and minimum of f occurs over the interval
[1.1,2]

Answer 1

The absolute maximum of f(x) = x³ / (x² - 1) over the interval [1.1, 2] occurs at x = 1.1, and the absolute minimum occurs at x = 1.5.

Step-by-step explanation:

To find the critical points of f(x) in the given interval, we first locate where the derivative is equal to zero or undefined. The derivative f'(x) can be calculated using the quotient rule:

f'(x) = (3x²(x² - 1) - x³(2x)) / (x² - 1)²

After finding f'(x), we set it equal to zero and solve for x:

(3x²(x² - 1) - x³(2x)) / (x² - 1)² = 0

Solving this equation gives us the critical points x = 1.1 and x = 1.5. We then evaluate f(x) at the critical points and endpoints of the interval:

f(1.1) = 1.1³ / (1.1² - 1) ≈ 0.602

f(1.5) = 1.5³ / (1.5² - 1) ≈ -1.286

f(2) = 2³ / (2² - 1) = 8 / 3

Comparing these values, we find that the absolute maximum occurs at x = 1.1, and the absolute minimum occurs at x = 1.5. Therefore, the final answer is as stated above.